In your use case, the actual norm of the vector does not matter since you are only concerned about the dominant eigenvalue. The only reason to normalize during the iteration is to keep the numbers from growing exponentially. You scale the vector however you want to prevent numeric overflow.
A key concept about eigenvectors and eigenvalues is that the set of vectors corresponding to an eigenvalue form a linear subspace. This is a consequence of multiplication by a matrix being a linear map. In particular, any scalar multiple of an eigenvector is also an eigenvector for the same eigenvalue.
The Wikipedia article Power method mentions the use of the Rayleigh quotient to compute an approximation to the dominant eigenvalue. For real vectors and matrices it is given by the value
$\, (v\cdot Av)/(v \cdot v). \,$ There are probably good reasons for the use of this formula. Of course, if $\,v\,$ is normalized so that $\, v \cdot v = 1, \,$ then you can simplify that to $\, v\cdot Av. \,$